Calculation of numeric output error values for velocity aberration correction of an image

ABSTRACT

When correcting for velocity aberration in satellite imagery, a closed-form error covariance propagation model can produce more easily calculable error terms than a corresponding Monte Carlo analysis. The closed-form error covariance propagation model is symbolic, rather than numeric. The symbolic error covariance propagation model relates input parameters to one another pairwise and in closed form. For a particular image, the symbolic error covariance propagation model receives an input measurement value and an input error value for each input parameter. The symbolic error covariance propagation model operates on the input values to produce a set of output correction values, which correct for velocity aberration. The output correction values can be used to convert apparent coordinate values to corrected coordinate values. The symbolic error covariance matrix operates on the input error values to produce a set of output error values, which identify a reliability of the corrected coordinate values.

TECHNICAL FIELD

Examples pertain generally to correcting registration errors insatellite imagery (errors in mapping from a point in an image to itscorresponding point on the earth's surface), and more particularly tocalculating error terms when correcting for velocity aberration insatellite images.

BACKGROUND

There are several near earth orbiting commercial satellites that canprovide images of structures or targets on the ground. For applicationsthat rely on these images, it is important to accurately register theimages to respective coordinates on the ground.

For satellite-generated images of earth-based targets, one known sourceof misregistration is referred to as velocity aberration. Velocityaberration can arise in an optical system with a sufficiently largevelocity relative to the point being imaged. A typical velocity of anear earth orbiting commercial satellite can be on the order of 7.5kilometers per second, with respect to a location on the earth directlybeneath the satellite. This velocity is large enough to produce aregistration error of several detector pixels at the satellite-basedcamera.

The correction for velocity aberration is generally well-known. However,it is generally challenging to calculate error terms associated with thecorrection. These error terms estimate the confidence level, orreliability, of the velocity aberration correction.

Historically, error calculation for velocity aberration correction hasbeen treated statistically with a Monte Carlo analysis. In general,these Monte Carlo analyses can be time-consuming and computationallyexpensive. In order to produce statistically significant results, aMonte Carlo analysis can require that a large number of simulated casesbe executed and analyzed, which can be difficult or impossible due tothe limitations of computational resources and processing timerequirements. As a result, error estimation for velocity aberrationcorrection can be lacking.

SUMMARY

When correcting for velocity aberration in satellite imagery, aclosed-form covariance matrix propagation can produce more reliable andmore easily calculable error terms than a corresponding matrix generatedby a Monte Carlo analysis. Performing calculations with the closed formcovariance matrix can be significantly faster than with a correspondingMonte Carlo analysis, can provide greater immunity to data outliers, andcan provide immediate checks for statistical consistency.

The closed-form covariance matrix propagation is symbolic, rather thannumeric. The symbolic covariance matrix propagation relates the knowncovariance matrix of the input parameters to the resulting covariancematrix of the aberration correction terms in closed form. For aparticular image, one employs the required input parameters to computethe velocity aberration correction terms at a selected point in theimage. The covariance matrix of the velocity aberration correction termsis then computed from the required input parameters, calculated velocityaberration terms, and the input parameter covariance matrix. Thenumerical values for the input parameter covariance matrix are receivedand the symbolic formulas are used to calculate the numerical values forthe covariance matrix of the velocity aberration correction terms.

The input and output error values are numerical values that indicate areliability, or confidence level, of a corresponding numericalcoordinate value or correction value. For instance, for a set ofcoordinates (x, y, z), the corresponding error values can be (σ_(x),σ_(y), σ_(z)). The value of σ_(x) represents the standard deviation ofx, while σ_(x) ² denotes the variance of x. In general, as the errorvalue σ_(x) decreases, the confidence in the reported value of xincreases.

This summary is intended to provide an overview of subject matter of thepresent patent application. It is not intended to provide an exclusiveor exhaustive explanation of the invention. The Detailed Description isincluded to provide further information about the present patentapplication.

BRIEF DESCRIPTION OF THE DRAWINGS

In the drawings, which are not necessarily drawn to scale, like numeralsmay describe similar components in different views. Like numerals havingdifferent letter suffixes may represent different instances of similarcomponents. The drawings illustrate generally, by way of example, butnot by way of limitation, various embodiments discussed in the presentdocument.

FIG. 1 is a schematic drawing of an example of a system for receivingand processing imagery, such as satellite imagery, in accordance withsome embodiments.

FIG. 2 is a flow chart of an example of a method for calculating numericoutput error values for velocity aberration correction of an image, inaccordance with some embodiments.

DETAILED DESCRIPTION

FIG. 1 is a schematic drawing of an example of a system 100 forreceiving and processing imagery, such as satellite imagery. A satellite102, such as an IKONOS or another near earth orbiting commercialsatellite, captures an image 104 of a target 106 on earth 108. Thesystem 100 downloads the captured image 104, along with associatedmetadata 110 corresponding to conditions under which the image 104 wastaken.

The metadata 110 includes a set of apparent coordinate values 112corresponding to a selected point in the image. The apparent coordinatevalues 112 suffer from velocity aberration. Velocity aberration isgenerally well-understood in the fields of astronomy and space-basedimaging. Velocity aberration produces a pointing error, so that theimage 104 formed at the sensor on the satellite is translated away fromits expected location. If velocity aberration is left uncorrected, theimage 104 and corresponding location on the target 106 can bemisregistered with respect to each other. Velocity aberration does notdegrade the image 104. The system 100 provides a correction for thevelocity aberration, and additionally provides a measure of reliabilityof the correction.

The metadata 110 also includes plurality of input values 114 forcorresponding input parameters 116. The input parameters 116 aregeometric quantities, such as distances and angular rates that definethe satellite sensor's position and velocity. The input parameters 116are defined symbolically and not numerically. The input values 114 arenumeric, with numerical values that correspond to the input parameters116.

Each input value 114 includes an input measured value 118, an input meanerror 120, and an input error standard deviation value 122. Each inputmeasured value 118 represents a measurement of a corresponding inputparameter 116 via onboard satellite instruments; this can be aconsidered a best estimate of the value of the input parameter. Eachinput mean error 120 represents an inherent bias of the correspondinginput measured value 118; if there is no bias, then the input mean error120 is zero. Each input error standard deviation value 122 represents areliability of the best estimate. A relatively low input error standarddeviation value 122 implies a relatively high confidence in thecorresponding input measured value 118, and a relatively high inputerror standard deviation value 122 implies a relatively low confidencein the corresponding input measured value 118. In some examples, atleast one mean input error standard deviation value 122 remainsinvariant for multiple images taken with a particular telescope in thesatellite 102. In some examples, all the input error standard deviationvalues 122 remain invariant for multiple images taken with a particulartelescope in the satellite 102.

A symbolic error covariance propagation model 124 receives the inputvalues 114 in the metadata 110. The symbolic error covariancepropagation model 124 includes a symbolic covariance matrix that relatesthe input parameters 116 to one another pairwise and in closed form. Thesymbolic covariance matrix can be image-independent, and can be used forother optical systems having the same configuration of input parameters116. The symbolic covariance matrix is symbolic, not numeric. In someexamples, the input parameters 116 and the symbolic covariance matrixremain invariant for multiple images taken with a particular telescopein the satellite 102. The Appendix to this document includes amathematical derivation of an example of a suitable symbolic covariancematrix.

Velocity aberration correction generates a set of output correctionvalues 126 from the required input values 114. The output correctionvalues 126 relate the apparent coordinate values 112 to a set ofcorrected coordinate values 128, and can therefore correct for velocityaberration in the image 104. The corrected coordinate values 128 can bestored, along with the image 104, and can be presented to a user as abest estimate of coordinates within the image 104.

The symbolic error covariance propagation model 124 generates a set ofoutput error values 130 from the input values 114. The output errorvalues 130 identify a reliability of the output correction values 126.The output error values 130 can also be stored, along with the image104, and can be presented to a user as a measure of reliability of thecoordinate correction.

As an example, a user of the system 100 can download an image 104 withmetadata 110. The system 100 can extract the suitable input values 114from the metadata 110. The system 100 can use extracted input values 114to calculate output correction values 126 to correct for velocityaberration in the image 104. The system 100 can apply the outputcorrection values 124 to a set of apparent coordinate values 112 togenerate a set of corrected coordinate values 128. The system 100 canuse the input values 114 to calculate output error values 130 thatestimate a confidence level or reliability of the corrected coordinatevalues 128. The system can present to the user the image 104, thecorrected coordinate values 128, and the output error values 130.

The system 100 can be a computer system that includes hardware, firmwareand software. Examples may also be implemented as instructions stored ona computer-readable storage device, which may be read and executed by atleast one processor to perform the operations described herein. Acomputer-readable storage device may include any non-transitorymechanism for storing information in a form readable by a machine (e.g.,a computer). For example, a computer-readable storage device may includeread-only memory (ROM), random-access memory (RAM), magnetic diskstorage media, optical storage media, flash-memory devices, and otherstorage devices and media. In some examples, computer systems caninclude one or more processors, optionally connected to a network, andmay be configured with instructions stored on a computer-readablestorage device.

The input parameters 116 can include a separation between a center ofmass of a telescope and a vertex of a primary mirror of the telescope,and include x, y, and z components of: a geocentric radius vector to acenter of mass of the telescope; a velocity vector to the center of massof the telescope; a geocentric radius vector to a target ground point;an angular rate vector of a body reference frame of the telescope; and aunit vector along a z-axis of the body reference frame of the telescope.These input parameters are but one example; other suitable inputparameters 116 can also be used.

FIG. 2 is a flow chart of an example of a method 200 for calculatingnumeric output error values for an image. The image has a correspondingset of output correction values that correct for velocity aberration.The method 200 can be executed by system 100 of FIG. 1, or by anothersuitable system.

At 202, method 200 receives metadata corresponding to conditions underwhich an image was taken. The metadata includes a plurality of inputvalues for corresponding input parameters. Each input value includes aninput measured value, an input mean error, and an input error standarddeviation value. At 204, method 200 provides the plurality of inputvalues to a symbolic error covariance propagation model. The symbolicerror covariance propagation model includes a symbolic covariance matrixthat relates the plurality of input parameters to one another pairwiseand in closed form. At 206, method 200 generates a set of output errorvalues from the symbolic error covariance propagation model and theplurality of input values. The set of output error values identifies areliability of the set of output correction values.

The remainder of the Detailed Description is an Appendix that includes amathematical derivation of an example of a symbolic error covariancepropagation model that is suitable for use in the system 100 of FIG. 1.The derived symbolic error covariance propagation model uses the inputparameters 116, as noted above.

APPENDIX

Beginning with the well-known Lorentz transformations from specialrelativity, one can relativistically formulate a true line of sightcorrection process, then use a first-order differential approximation toconstruct a covariance propagation model of the aberration correctionprocess. The first-order differential approximation acknowledges thatrelative velocities between reference frames are significantly less thanthe speed of light, which is the case for near earth orbiting commercialsatellites.

There are sixteen input parameters employed in the calculation of thecorrected line-of-sight. Each input parameter has an input measuredvalue (X), an input mean error value (μ_(X)), and an input error term(σ_(X)).

The sixteen input error terms are defined symbolically as follows.Quantities Δx_(cm), Δy_(cm), Δz_(cm) are three random error componentsof a geocentric radius vector

_(CM) to a telescope center of mass. Quantities Δv_(xcm), Δv_(ycm),Δv_(zcm) are three random error components of a velocity vector

_(CM) to the telescope center of mass. Quantities Δx_(p), Δy_(p), Δz_(p)are three random error components of a geocentric radius vector

_(P) to a target ground point. Quantity Δd_(Z) is a random errorcomponent of a distance from a center of mass to the vertex of thetelescope's primary mirror. Quantities Δω_(xb), Δω_(yb), Δω_(zb) arethree random error components of an angular rate vector

_(B) of the telescope's body reference frame. Quantities Δz_(xb),Δz_(yb), Δz_(zb) are three random error components of a unit vector{circumflex over (Z)}_(B) along a Z axis of the telescope's body frame.In some examples, one or more of the sixteen input error terms can beomitted.

Quantity Δ{circumflex over (q)}′_(3×1) is an error vector for corrected(true) line-of-sight components:

${\Delta \; {\hat{q}}_{3 \times 1}^{\prime}} = \begin{pmatrix}{\Delta \; x_{true}} \\{\Delta \; y_{true}} \\{\Delta \; z_{true}}\end{pmatrix}$

Quantity H_(3×16) is a matrix of partial derivations of the threecomponents of the error vector Δ{circumflex over (q)}′_(3×1), withrespect to the sixteen input error terms:

$H_{3 \times 16} = \begin{pmatrix}\frac{\partial q_{x}^{\prime}}{\partial x_{c\; m}} & \frac{\partial q_{x}^{\prime}}{\partial y_{c\; m}} & \frac{\partial q_{x}^{\prime}}{\partial z_{c\; m}} & \frac{\partial q_{x}^{\prime}}{\partial v_{xcm}} & \ldots & \frac{\partial q_{x}^{\prime}}{\partial z_{zb}} \\\frac{\partial q_{y}^{\prime}}{\partial x_{c\; m}} & \frac{\partial q_{y}^{\prime}}{\partial y_{c\; m}} & \frac{\partial q_{y}^{\prime}}{\partial z_{c\; m}} & \frac{\partial q_{y}^{\prime}}{\partial v_{xcm}} & \ldots & \frac{\partial q_{y}^{\prime}}{\partial z_{zb}} \\\frac{\partial q_{z}^{\prime}}{\partial x_{c\; m}} & \frac{\partial q_{z}^{\prime}}{\partial y_{c\; m}} & \frac{\partial q_{z}^{\prime}}{\partial z_{c\; m}} & \frac{\partial q_{z}^{\prime}}{\partial v_{xcm}} & \ldots & \frac{\partial q_{z}^{\prime}}{\partial z_{zb}}\end{pmatrix}$

Quantity Δ

_(16×1) is an input error vector, formed from the sixteen input errorterms:

${\Delta \; {\overset{\rightharpoonup}{\Psi}}_{16 \times 1}} = \begin{pmatrix}{\Delta \; x_{c\; m}} \\{\Delta \; y_{c\; m}} \\{\Delta \; z_{c\; m}} \\{\Delta \; v_{xcm}} \\{\Delta \; v_{ycm}} \\{\Delta \; v_{zcm}} \\{\Delta \; x_{p}} \\{\Delta \; y_{p}} \\{\Delta \; z_{p}} \\{\Delta \; d_{Z}} \\{\Delta \; \omega_{xb}} \\{\Delta \; \omega_{yb}} \\{\Delta \; \omega_{zb}} \\{\Delta \; z_{xb}} \\{\Delta \; z_{yb}} \\{\Delta \; z_{{zb}\;}}\end{pmatrix}$

A first-order differential approximation can linearly relate thequantities Δ{circumflex over (q)}′_(3×1), H_(3×16), and Δ

_(16×1) to one other:

Δ{circumflex over (q)}′ _(3×1) =H _(3×16)Δ

_(16×1)

Applying an expectation operator leads to:

Δ{circumflex over (q)}′ _(3×1)

=H _(3×16)

Δ

_(16×1)

,  (1)

Quantity P_(Δ{circumflex over (q)}′) is a covarianceP_(Δ{circumflex over (q)}′) of resulting errors in the calculatedline-of-sight components, and can be calculated by:

P _(Δ{circumflex over (q)}′)≡

(Δ{circumflex over (q)}′−

Δ{circumflex over (q)}′

)(Δ{circumflex over (q)}′−

Δ{circumflex over (q)}′

)^(T)

P _(Δ{circumflex over (q)}′) =H

(Δ

−

Δ

)(Δ

−

Δ

)^(T)

H ^(T)

P _(Δ{circumflex over (q)}′) =H

H ^(T),  (2)

where

is a 16-by-16 covariance matrix of input parameter errors. Equations (1)and (2) form a covariance propagation model. The covariance propagationmodel relates the means and covariance of the random measurement errorsin the parameters input into the aberration correction process to themeans and covariance of the components of the corrected true line ofsight.

The preceding is but one example of a symbolic error covariancepropagation model; other suitable symbolic error covariance propagationmodels can also be used.

The indicated partial derivatives that constitute the matrix H_(3×16)are now evaluated in detail. The first step is to give detailedexpressions for q′_(x), q′_(y), q′_(z):

$\begin{matrix}{q_{x}^{\prime} = {{\left( \frac{\left( {\hat{q} \cdot \hat{u}} \right)\sqrt{1 - \beta^{2}}}{1 - {\beta \left( {\hat{q} \cdot \hat{v}} \right)}} \right)\left( {\hat{u} \cdot \hat{i}} \right)} + {\left( \frac{\left( {\hat{q} \cdot \hat{v}} \right) - \beta}{1 - {\beta \left( {\hat{q} \cdot \hat{v}} \right)}} \right)\left( {\hat{v} \cdot \hat{i}} \right)}}} & (3) \\{q_{y}^{\prime} = {{\left( \frac{\left( {\hat{q} \cdot \hat{u}} \right)\sqrt{1 - \beta^{2}}}{1 - {\beta \left( {\hat{q} \cdot \hat{v}} \right)}} \right)\left( {\hat{u} \cdot \hat{j}} \right)} + {\left( \frac{\left( {\hat{q} \cdot \hat{v}} \right) - \beta}{1 - {\beta \left( {\hat{q} \cdot \hat{v}} \right)}} \right)\left( {\hat{v} \cdot \hat{j}} \right)}}} & (4) \\{q_{z}^{\prime} = {{\left( \frac{\left( {\hat{q} \cdot \hat{u}} \right)\sqrt{1 - \beta^{2}}}{1 - {\beta \left( {\hat{q} \cdot \hat{v}} \right)}} \right)\left( {\hat{u} \cdot \hat{k}} \right)} + {\left( \frac{\left( {\hat{q} \cdot \hat{v}} \right) - \beta}{1 - {\beta \left( {\hat{q} \cdot \hat{v}} \right)}} \right)\left( {\hat{v} \cdot \hat{k}} \right)}}} & (5)\end{matrix}$

Let ξ represent any of the sixteen scalar input parameters. Directdifferentiation of equations 3, 4, and 5 yields:

$\begin{matrix}{\frac{\partial q_{x}^{\prime}}{\partial\xi} = {{\left\lbrack \frac{\begin{matrix}{{\left\lbrack {\left( {\frac{\partial\hat{q}}{\partial\xi} \cdot \hat{u}} \right) + \left( {\hat{q} \cdot \frac{\partial\hat{u}}{\partial\xi}} \right)} \right\rbrack \sqrt{1 - \beta^{2}}} -} \\{\frac{\left( {\hat{q} - \hat{u}} \right)}{\sqrt{1 - \beta^{2}}}\beta \frac{\partial\beta}{\partial\xi}}\end{matrix}}{1 - {\beta \left( {\hat{q} \cdot \hat{v}} \right)}} \right\rbrack \left( {\hat{u} \cdot \hat{i}} \right)} + {{\left\lbrack \frac{\left( {\hat{q} \cdot \hat{v}} \right)\sqrt{1 - \beta^{2}}}{\left\{ {1 - {\beta \left( {\hat{q} \cdot \hat{v}} \right)}} \right\}^{2}} \right\rbrack \begin{bmatrix}{{\frac{\partial\beta}{\partial\xi}\left( {\hat{q} \cdot \hat{v}} \right)} +} \\{\beta \left\lbrack {\left( {\frac{\partial\hat{q}}{\partial\xi} \cdot \hat{v}} \right) + \left( {\hat{q} \cdot \frac{\partial\hat{v}}{\partial\xi}} \right)} \right\rbrack}\end{bmatrix}}\left( {\hat{u} \cdot \hat{i}} \right)} + {\left( \frac{\left( {\hat{q} \cdot \hat{v}} \right)\sqrt{1 - \beta^{2}}}{1 - {\beta \left( {\hat{q} \cdot \hat{v}} \right)}} \right)\left\lbrack \left( {\frac{\partial\hat{u}}{\partial\xi} \cdot \hat{i}} \right) \right\rbrack} + {\left\lbrack \frac{\left\lbrack {\left( {\frac{\partial\hat{q}}{\partial\xi} \cdot \hat{v}} \right) + \left( {\hat{q} \cdot \frac{\partial\hat{v}}{\partial\xi}} \right)} \right\rbrack - \frac{\partial\beta}{\partial\xi}}{1 - {\beta \left( {\hat{q} \cdot \hat{v}} \right)}} \right\rbrack \left( {\hat{v} \cdot \hat{i}} \right)} + {{\frac{\left( {\left( {\hat{q} \cdot \hat{v}} \right) - \beta} \right)}{\left( {1 - {\beta \left( {\hat{q} \cdot \hat{v}} \right)}} \right)^{2}}\left\lbrack {{\frac{\partial\beta}{\partial\xi}\left( {\hat{q} \cdot \hat{v}} \right)} + {\beta \left\{ {\left( {\frac{\partial\hat{q}}{\partial\xi} \cdot \hat{v}} \right) + \left( {\hat{q} \cdot \frac{\partial\hat{v}}{\partial\xi}} \right)} \right\}}} \right\rbrack}\left( {\hat{v} \cdot \hat{i}} \right)} + {\left( \frac{\left( {\hat{q} \cdot \hat{v}} \right) - \beta}{1 - {\beta \left( {\hat{q} \cdot \hat{v}} \right)}} \right)\left\lbrack \left( {\frac{\partial\hat{v}}{\partial\xi} \cdot \hat{i}} \right) \right\rbrack}}} & (6) \\{\frac{\partial q_{y}^{\prime}}{\partial\xi} = {{\left\lbrack \frac{\begin{matrix}{{\left\lbrack {\left( {\frac{\partial\hat{q}}{\partial\xi} \cdot \hat{u}} \right) + \left( {\hat{q} \cdot \frac{\partial\hat{u}}{\partial\xi}} \right)} \right\rbrack \sqrt{1 - \beta^{2\;}}} -} \\{\frac{\left( {\hat{q} \cdot \hat{u}} \right)}{\sqrt{1 - \beta^{2}}}\beta \; \frac{\partial\beta}{\partial\xi}}\end{matrix}}{1 - {\beta \left( {\hat{q} \cdot \hat{v}} \right)}} \right\rbrack \left( {\hat{u} \cdot \hat{j}} \right)} + {{\left\lbrack \frac{\left( {\hat{q} \cdot \hat{u}} \right)\sqrt{1 - \beta^{2}}}{\left\{ {1 - {\beta \left( {\hat{q} \cdot \hat{v}} \right)}} \right\}^{2}} \right\rbrack \begin{bmatrix}{{\frac{\partial\beta}{\partial\xi}\left( {\hat{q} \cdot \hat{v}} \right)} +} \\{\beta \left\lbrack {\left( {\frac{\partial\hat{q}}{\partial\xi} \cdot \hat{v}} \right) + \left( {\hat{q} \cdot \frac{\partial\hat{v}}{\partial\xi}} \right)} \right\rbrack}\end{bmatrix}}\left( {\hat{u} \cdot \hat{j}} \right)} + {\left( \frac{\left( {\hat{q} \cdot \hat{u}} \right)\sqrt{1 - \beta^{2}}}{1 - {\beta \left( {\hat{q} \cdot \hat{v}} \right)}} \right)\left\lbrack \left( {\frac{\partial\hat{u}}{\partial\xi} \cdot \hat{j}} \right) \right\rbrack} + {\left\lbrack \frac{\left\lbrack {\left( {\frac{\partial\hat{q}}{\partial\xi} \cdot \hat{v}} \right) + \left( {q \cdot \frac{\partial\hat{v}}{\partial\xi}} \right)} \right\rbrack - \frac{\partial\beta}{\partial\xi}}{1 - {\beta \left( {\hat{q} \cdot \hat{v}} \right)}} \right\rbrack \left( {\hat{v} \cdot \hat{j}} \right)} + {{\frac{\left( {\left( {\hat{q} \cdot \hat{v}} \right) - \beta} \right)}{\left( {1 - {\beta \left( {\hat{q} \cdot \hat{v}} \right)}} \right)^{2}}\left\lbrack {{\frac{\partial\beta}{\partial\xi}\left( {\hat{q} \cdot \hat{v}} \right)} + {\beta \left\{ {\left( {\frac{\partial\hat{q}}{\partial\xi} \cdot \hat{v}} \right) + \left( {\hat{q} \cdot \frac{\partial\hat{v}}{\partial\xi}} \right)} \right\}}} \right\rbrack}\left( {\hat{v} \cdot \hat{j}} \right)} + {\left( \frac{\left( {\hat{q} \cdot \hat{v}} \right) - \beta}{1 - {\beta \left( {\hat{q} \cdot \hat{v}} \right)}} \right)\left\lbrack \left( {\frac{\partial\hat{v}}{\partial\xi} \cdot \hat{j}} \right) \right\rbrack}}} & (7) \\{\frac{\partial q_{z}^{\prime}}{\partial\xi} = {{\left\lbrack \frac{\begin{matrix}{{\left\lbrack {\left( {\frac{\partial\hat{q}}{\partial\xi} \cdot \hat{u}} \right) + \left( {\hat{q} \cdot \frac{\partial\hat{u}}{\partial\xi}} \right)} \right\rbrack \sqrt{1 - \beta^{2}}} -} \\{\frac{\left( {\hat{q} \cdot \hat{u}} \right)}{\sqrt{1 - \beta^{2}}}\beta \; \frac{\partial\beta}{\partial\xi}}\end{matrix}}{1 - {\beta \left( {\hat{q} \cdot \hat{v}} \right)}} \right\rbrack \left( {\hat{u} \cdot \hat{k}} \right)} + {{\left\lbrack \frac{\left( {\hat{q} \cdot \hat{u}} \right)\sqrt{1 - \beta^{2\;}}}{\left\{ {1 - {\beta \left( {\hat{q} \cdot \hat{v}} \right)}} \right\}^{2}} \right\rbrack \begin{bmatrix}{{\frac{\partial\beta}{\partial\xi}\left( {\hat{q} \cdot \hat{v}} \right)} +} \\{\beta \left\lbrack {\left( {\frac{\partial\hat{q}}{\partial\xi} \cdot \hat{v}} \right) + \left( {\hat{q} \cdot \frac{\partial\hat{v}}{\partial\xi}} \right)} \right\rbrack}\end{bmatrix}}\left( {\hat{u} \cdot \hat{k}} \right)} + {\left( \frac{\left( {\hat{q} \cdot \hat{u}} \right)\sqrt{1 - \beta^{2}}}{1 - {\beta \left( {\hat{q} \cdot \hat{v}} \right)}} \right)\left\lbrack \left( {\frac{\partial\hat{u}}{\partial\xi} \cdot \hat{k}} \right) \right\rbrack} + {\left\lbrack \frac{\left\lbrack {\left( {\frac{\partial\hat{q}}{\partial\xi} \cdot \hat{v}} \right) + \left( {\hat{q} \cdot \frac{\partial\hat{v}}{\partial\xi}} \right)} \right\rbrack - \frac{\partial\beta}{\partial\xi}}{1 - {\beta \left( {\hat{q} \cdot \hat{v}} \right)}} \right\rbrack \left( {\hat{v} \cdot \hat{k}} \right)} + {{\frac{\left( {\left( {\hat{q} \cdot \hat{v}} \right) - \beta} \right)}{\left( {1 - {\beta \left( {\hat{q} \cdot \hat{v}} \right)}} \right)^{2}}\left\lbrack {{\frac{\partial\beta}{\partial\xi}\left( {\hat{q} \cdot \hat{v}} \right)} + {\beta \left\{ {\left( {\frac{\partial\hat{q}}{\partial\xi} \cdot \hat{v}} \right) + \left( {\hat{q} \cdot \frac{\partial\hat{v}}{\partial\xi}} \right)} \right\}}} \right\rbrack}\left( {\hat{v} \cdot \hat{k}} \right)} + {\left( \frac{\left( {\hat{q} \cdot \hat{v}} \right) - \beta}{1 - {\beta \left( {\hat{q} \cdot \hat{v}} \right)}} \right)\left\lbrack \left( {\frac{\partial\hat{v}}{\partial\xi} \cdot \hat{k}} \right) \right\rbrack}}} & (8)\end{matrix}$

The next indicated partial derivatives to be evaluated are:

$\frac{\partial\beta}{\partial\xi},\frac{\partial\hat{u}}{\partial\xi},\frac{\partial\hat{v}}{\partial\xi},{\frac{\partial\hat{q}}{\partial\xi}.}$

Direct differentiation yields:

$\begin{matrix}{\frac{\partial\beta}{\partial\xi} = \frac{\begin{Bmatrix}{\frac{\partial{\overset{\overset{.}{\rightharpoonup}}{R}}_{CM}}{\partial\xi} + {\frac{\partial d_{Z}}{\partial\xi}\left( {{\overset{\rightharpoonup}{\omega}}_{B} \times {\hat{Z}}_{B}} \right)} +} \\{d_{Z}\left( {{\frac{\partial{\overset{\rightharpoonup}{\omega}}_{B}}{\partial\xi} \times {\overset{\rightharpoonup}{Z}}_{B}} + {{\overset{\rightharpoonup}{\omega}}_{B} \times \frac{\partial{\overset{\rightharpoonup}{Z}}_{B}}{\partial\xi}}} \right)}\end{Bmatrix} \cdot \left\lbrack {{\overset{\overset{.}{\rightharpoonup}}{R}}_{CM} + {d_{Z}\left( {{\overset{\rightharpoonup}{\omega}}_{B} \times {\hat{Z}}_{B}} \right)}} \right\rbrack}{2\sqrt{\left\lbrack {{- {\overset{\overset{.}{\rightharpoonup}}{R}}_{CM}} - {d_{Z}\left( {{\overset{\rightharpoonup}{\omega}}_{B} \times {\hat{Z}}_{B}} \right)}} \right\rbrack \cdot \left\lbrack {{- {\overset{\overset{.}{\rightharpoonup}}{R}}_{CM}} - {d_{Z}\left( {{\overset{\rightharpoonup}{\omega}}_{B} \times {\hat{Z}}_{B}} \right)}} \right\rbrack}}} & (9) \\{\frac{\partial\hat{u}}{\partial\xi} = {\frac{\frac{\partial\hat{q}}{\partial\xi} - {\left( {{\frac{\partial\hat{q}}{\partial\xi} \cdot \hat{v}} + {\hat{q} \cdot \frac{\partial\hat{v}}{\partial\xi}}} \right)\hat{v}} + {\left( {\hat{q} \cdot \hat{v}} \right)\frac{\partial\hat{v}}{\partial\xi}}}{\sqrt{\left( {\hat{q} - {\left( {\hat{q} \cdot \hat{v}} \right)\hat{v}}} \right) \cdot \left( {\hat{q} - {\left( {\hat{q} \cdot \hat{v}} \right)\hat{v}}} \right)}} - {\left( {\hat{q} - {\left( {\hat{q} \cdot \hat{v}} \right)\hat{v}}} \right)\frac{\left\{ {\frac{\partial\hat{q}}{\partial\xi} - {\left( {{\frac{\partial\hat{q}}{\partial\xi} \cdot \hat{v}} + {\hat{q} \cdot \frac{\partial\hat{v}}{\partial\xi}}} \right)\hat{v}} + {\left( {\hat{q} \cdot \hat{v}} \right)\frac{\partial\hat{v}}{\partial\xi}}} \right\} \cdot \left( {\hat{q} - {\left( {\hat{q} \cdot \hat{v}} \right)\hat{v}}} \right)}{\left\lbrack {\left( {\hat{q} - {\left( {\hat{q} \cdot \hat{v}} \right)\hat{v}}} \right) \cdot \left( {\hat{q} - {\left( {\hat{q} \cdot \hat{v}} \right)\hat{v}}} \right)} \right\rbrack^{\frac{3}{2}}}}}} & (10) \\{\frac{\partial\hat{v}}{\partial\xi} = {\frac{{- \frac{\partial{\overset{\overset{.}{\rightharpoonup}}{R}}_{CM}}{\partial\xi}} - {\frac{\partial d_{Z}}{\partial\xi}\left( {{\overset{\rightharpoonup}{\omega}}_{B} \times {\hat{Z}}_{B}} \right)} - {d_{Z}\left( {{\frac{\partial{\overset{\rightharpoonup}{\omega}}_{B}}{\partial\xi} \times {\hat{Z}}_{B}} + {{\overset{\rightharpoonup}{\omega}}_{B} \times \frac{\partial{\hat{Z}}_{B}}{\partial\xi}}} \right)}}{\sqrt{{- \left( {{\overset{\overset{.}{\rightharpoonup}}{R}}_{CM} - {d_{Z}\left( {{\overset{\rightharpoonup}{\omega}}_{B} \times {\hat{Z}}_{B}} \right)}} \right)} \cdot \left( {{- {\overset{\overset{.}{\rightharpoonup}}{R}}_{CM}} - {d_{Z}\left( {{\overset{\rightharpoonup}{\omega}}_{B} \times {\hat{Z}}_{B}} \right)}} \right)}} + \frac{\begin{matrix}\left( {{\overset{\overset{.}{\rightharpoonup}}{R}}_{CM} + {d_{Z}\left( {{\overset{\rightharpoonup}{\omega}}_{B} \times {\hat{Z}}_{B}} \right)}} \right) \\\left\{ {\begin{bmatrix}{\frac{\partial{\overset{\overset{.}{\rightharpoonup}}{R}}_{CM}}{\partial\xi} + {\frac{\partial d_{Z}}{\partial\xi}\left( {{\overset{\rightharpoonup}{\omega}}_{B} \times {\hat{Z}}_{B}} \right)} +} \\{d_{Z}\left( {{\frac{\partial{\overset{\rightharpoonup}{\omega}}_{B}}{\partial\xi} \times {\hat{Z}}_{B}} + {{\overset{\rightharpoonup}{\omega}}_{B} \times \frac{\partial{\hat{Z}}_{B}}{\partial\xi}}} \right)}\end{bmatrix} \cdot \left( {{\overset{\overset{.}{\rightharpoonup}}{R}}_{CM} + {d_{Z}\left( {{\overset{\rightharpoonup}{\omega}}_{B} \times {\hat{Z}}_{B}} \right)}} \right)} \right\}\end{matrix}}{\left\{ {\left( {{- {\overset{\overset{.}{\rightharpoonup}}{R}}_{CM}} - {d_{Z}\left( {{\overset{\rightharpoonup}{\omega}}_{B} \times {\hat{Z}}_{B}} \right)}} \right) \cdot \left( {{- {\overset{\overset{.}{\rightharpoonup}}{R}}_{CM}} - {d_{Z}\left( {{\overset{\rightharpoonup}{\omega}}_{B} \times {\hat{Z}}_{B}} \right)}} \right)} \right\}^{\frac{3}{2}}}}} & (11) \\{\frac{\partial\hat{q}}{\partial\xi} = {\frac{\frac{\partial{\overset{\rightharpoonup}{R}}_{P}}{\partial\xi} - {\frac{\partial d_{Z}}{\partial\xi}{\hat{Z}}_{B}} - {d_{Z}\frac{\partial{\hat{Z}}_{B}}{\partial\xi}} - \frac{\partial{\overset{\rightharpoonup}{R}}_{CM}}{\partial\xi}}{\sqrt{\left( {{\overset{\rightharpoonup}{R}}_{P} - {d_{Z}{\hat{Z}}_{B}} - {\overset{\rightharpoonup}{R}}_{CM}} \right) \cdot \left( {{\overset{\rightharpoonup}{R}}_{P} - {d_{Z}{\hat{Z}}_{B}} - {\overset{\rightharpoonup}{R}}_{CM}} \right)}} - {\left( {{\overset{\rightharpoonup}{R}}_{P} - {d_{Z}{\hat{Z}}_{B}} - {\overset{\rightharpoonup}{R}}_{CM}} \right)\frac{\begin{Bmatrix}{\left( {\frac{\partial{\overset{\rightharpoonup}{R}}_{P}}{\partial\xi} - {\frac{\partial d_{Z}}{\partial\xi}{\hat{Z}}_{B}} - {d_{Z}\frac{\partial{\hat{Z}}_{B}}{\partial\xi}} - \frac{\partial{\overset{\rightharpoonup}{R}}_{CM}}{\partial\xi}} \right) \cdot} \\\left( {{\overset{\rightharpoonup}{R}}_{P} - {d_{Z}{\hat{Z}}_{B}} - {\overset{\rightharpoonup}{R}}_{CM}} \right)\end{Bmatrix}}{\left\lbrack {\left( {{\overset{\rightharpoonup}{R}}_{P} - {d_{Z}{\hat{Z}}_{B}} - {\overset{\rightharpoonup}{R}}_{CM}} \right) \cdot \left( {{\overset{\rightharpoonup}{R}}_{P} - {d_{Z}{\hat{Z}}_{B}} - {\overset{\rightharpoonup}{R}}_{CM}} \right)} \right\rbrack^{\frac{3}{2}}}}}} & (12)\end{matrix}$

The remaining partial derivatives to be evaluated are:

$\frac{\partial{\overset{\rightharpoonup}{R}}_{CM}}{\partial\xi},\frac{\partial{\overset{\overset{.}{\rightharpoonup}}{R}}_{CM}}{\partial\xi},\frac{\partial{\overset{\rightharpoonup}{R}}_{P}}{\partial\xi},\frac{\partial d_{Z}}{\partial\xi},\frac{\partial{\overset{\rightharpoonup}{\omega}}_{B}}{\partial\xi},{\frac{\partial{\hat{Z}}_{B}}{\partial\xi}.}$

Given the possible values that the dummy variable ξ may assume, onlysixteen of the partial derivatives above are non-zero. These partialderivatives are:

$\begin{matrix}{\frac{\partial{\overset{\rightharpoonup}{R}}_{CM}}{\partial x_{c\; m}} = {\frac{\partial{\overset{\overset{.}{\rightharpoonup}}{R}}_{CM}}{\partial v_{xcm}} = {\frac{\partial{\overset{\rightharpoonup}{R}}_{P}}{\partial x_{p}} = {\frac{\partial{\overset{\rightharpoonup}{\omega}}_{B}}{\partial\omega_{xb}} = {\frac{\partial{\hat{Z}}_{B}}{\partial Z_{xb}} = \hat{i}}}}}} & (13) \\{\frac{\partial{\overset{\rightharpoonup}{R}}_{CM}}{\partial y_{c\; m}} = {\frac{\partial{\overset{\overset{.}{\rightharpoonup}}{R}}_{CM}}{\partial v_{ycm}} = {\frac{\partial{\overset{\rightharpoonup}{R}}_{P}}{\partial y_{p}} = {\frac{\partial{\overset{\rightharpoonup}{\omega}}_{B}}{\partial\omega_{yb}} = {\frac{\partial{\hat{Z}}_{B}}{\partial Z_{yb}} = \hat{j}}}}}} & (14) \\{\frac{\partial{\overset{\rightharpoonup}{R}}_{CM}}{\partial z_{c\; m}} = {\frac{\partial{\overset{\overset{.}{\rightharpoonup}}{R}}_{CM}}{\partial v_{zcm}} = {\frac{\partial{\overset{\rightharpoonup}{R}}_{P}}{\partial z_{p}} = {\frac{\partial{\overset{\rightharpoonup}{\omega}}_{B}}{\partial\omega_{zb}} = {\frac{\partial{\hat{Z}}_{B}}{\partial Z_{{zb}\;}} = \hat{k}}}}}} & (15) \\{\frac{\partial d_{Z}}{\partial d_{Z}} = 1} & (16)\end{matrix}$

This constitutes all the partial derivatives needed to evaluate thematrix H_(3×16).

What is claimed is:
 1. A method for calculating numeric output errorvalues for an image, the image having a corresponding set of outputcorrection values that correct for velocity aberration, the methodcomprising: receiving metadata corresponding to conditions under whichthe image was taken, the metadata including a plurality of input valuesfor corresponding input parameters, each input value including an inputmeasured value, an input mean error, and an input error standarddeviation value; providing the plurality of input values to a symbolicerror covariance propagation model, the symbolic error covariancepropagation model including a symbolic covariance matrix that relatesthe plurality of input parameters to one another pairwise and in closedform; and generating a set of output error values from the symbolicerror covariance propagation model and the plurality of input values,the set of output error values identifying a reliability of the set ofoutput correction values.
 2. The method of claim 1, wherein the inputparameters include a separation between a center of mass of a telescopeand a vertex of a primary mirror of the telescope, and include x, y, andz components of: a geocentric radius vector to a center of mass of thetelescope; a velocity vector to the center of mass of the telescope; ageocentric radius vector to a target ground point; an angular ratevector of a body reference frame of the telescope; and a unit vectoralong a z-axis of the body reference frame of the telescope.
 3. Themethod of claim 1, wherein each input measured value represents a bestestimate of a corresponding input parameter.
 4. The method of claim 3,wherein each an input error standard deviation value represents areliability of the best estimate.
 5. The method of claim 1, wherein oneor more input error standard deviation values remains invariant formultiple images taken with a particular telescope.
 6. The method ofclaim 1, further comprising generating the set of output correctionvalues from the plurality of input measured values.
 7. A system forprocessing imagery, comprising: a processor; and memory, coupled to theprocessor, for storing an image, storing metadata corresponding toconditions under which the image was taken, and storing a correspondingset of output correction values that correct for velocity aberration;wherein the processor is configured to: receive the metadata, themetadata including a plurality of input values for corresponding inputparameters, each input value including an input measured value, an inputmean error, and an input error standard deviation value; provide theplurality of input values to a symbolic error covariance propagationmodel, the symbolic error covariance propagation model including asymbolic covariance matrix that relates the plurality of inputparameters to one another pairwise and in closed form; and generate aset of output error values from the symbolic error covariancepropagation model and the plurality of input values, the set of outputerror values identifying a reliability of the set of output correctionvalues.
 8. The system of claim 7, wherein the input parameters include aseparation between a center of mass of a telescope and a vertex of aprimary mirror of the telescope, and include x, y, and z components of:a geocentric radius vector to a center of mass of the telescope; avelocity vector to the center of mass of the telescope; a geocentricradius vector to a target ground point; an angular rate vector of a bodyreference frame of the telescope; and a unit vector along a z-axis ofthe body reference frame of the telescope.
 9. The system of claim 7,wherein each input measured value represents a best estimate of acorresponding input parameter.
 10. The system of claim 9, wherein eachinput error standard deviation value represents a reliability of thebest estimate.
 11. The system of claim 7, wherein one or more inputerror standard deviation values remains invariant for multiple imagestaken with a particular telescope.
 12. The system of claim 7, furthercomprising generating the set of output correction values from theplurality of input measured values.
 13. A computer-readable storagemedium storing a program for causing a computer to implement a methodfor calculating numeric output error values for an image, the imagehaving a corresponding set of output correction values that correct forvelocity aberration, the method comprising: receiving metadatacorresponding to conditions under which the image was taken, themetadata including a plurality of input values for corresponding inputparameters, each input value including an input measured value, an inputmean error, and an input error standard deviation value; providing theplurality of input values to a symbolic error covariance propagationmodel, the symbolic error covariance propagation model including asymbolic covariance matrix that relates the plurality of inputparameters to one another pairwise and in closed form; and generating aset of output error values from the symbolic error covariancepropagation model and the plurality of input values, the set of outputerror values identifying a reliability of the set of output correctionvalues.
 14. The computer-readable storage medium of claim 13, whereinthe input parameters include a separation between a center of mass of atelescope and a vertex of a primary mirror of the telescope, and includex, y, and z components of: a geocentric radius vector to a center ofmass of the telescope; a velocity vector to the center of mass of thetelescope; a geocentric radius vector to a target ground point; anangular rate vector of a body reference frame of the telescope; and aunit vector along a z-axis of the body reference frame of the telescope.15. The computer-readable storage medium of claim 13, wherein each inputmeasured value represents a best estimate of a corresponding inputparameter.
 16. The computer-readable storage medium of claim 15, whereineach an input error standard deviation value represents a reliability ofthe best estimate.
 17. The computer-readable storage medium of claim 13,wherein one or more input error standard deviation values remainsinvariant for multiple images taken with a particular telescope.
 18. Thecomputer-readable storage medium of claim 13, further comprisinggenerating the set of output correction values from the plurality ofinput measured values.